In many systems, there is a need to process a signal non-linearly, so that the signal stays within certain (constant or signal-dependent) boundaries. It is often desirable that the signal also is kept within a certain bandwidth. In particular in radio signal applications, this ensures that it does not spill over into adjacent channels or exceeds spectral emission limits.
When non-linear processing and filtering force a time-discrete signal to stay within certain boundaries, this can generally only be guaranteed at the sample instants. As the time-discrete signal is converted into time-continuous form, or upsampled to a higher sampling rate, it will typically exhibit overshoots that go outside the set boundaries, sometimes far outside. Generally, lower oversampling ratio (OSR) and sharper non-linear processing give rise to larger overshoots that are seen after upsampling or conversion to time-continuous form.
The traditional solution to this problem is to perform the non-linear/filter processing at a sufficiently high rate from the start. The problem with such solution is that processing at a higher sample rate (higher OSR) costs more in terms of operations per second, amount of hardware, or power consumption.